Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Abstract

We study the cubic defocusing nonlinear Schrödinger equation on R 4 with supercritical initial data. For randomized initial data in H s (R 4 ), we prove almost sure local wellposedness for 1 7 < s < 1 and almost sure scattering for 5 7 < s < 1. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and -for the almost sure scattering result -an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, where we effectively combine the advantages of the different decompositions.

Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

“…Previously, there is no probabilistic global result for the 3D energy critical NLS with large data. As mentioned above, for the 4D case, the almost sure scattering in the radial case was proved in [26,30] and extended to the non-radial case in [39] very recently. All the above 4D results consider the initial data in H s…”

“…The related results on non-radial energy-critical nonlinear Klein-Gordon equations were studied by Chen and Wang [19] for d = 4 and 5. In the very recently, we also learn that the almost sure scattering for the 4D non-radial NLS was obtained by Spitz [39] in H s x with 5 7 < s < 1, using a randomization similar to the one introduced by Burq and Krieger [13].…”

On the almost sure scattering for the energy-critical cubic wave equation with supercritical data

“…We note that in [10] a different randomization than in [22,21] for radially symmetric data was used. In fact, most of the aforementioned results (except [9,40,38]) relied on the so-called Wiener randomization, which is based on a unit-scale decomposition of frequency space (see Subsection 1.2 below for details).…”

“…. We use the overall strategy developed in [22] and successfully applied in [25,21,10,40] to obtain almost sure scattering results for the energy-critical cubic nonlinear wave and Schrödinger equation. The strategy consists in developing a suitable local wellposedness and perturbation theory for the forced equation (1.19), which is then combined with the existing deterministic theory for (1.1) to derive a scattering result for (1.19) conditioned on an a priori bound of the energy E(v) of v. Note that the energy of v is not conserved as v is not a solution of the energy-critical NLW (1.1).…”

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

“…In the following, we use a randomization of initial data based on a unit-scale decomposition of frequency space. Let us mention that other randomizations have been investigated in [BTT13,Den12,Tho09,Spi21]. Let ψ ∈ C ∞ c (R d ) be an even, non-negative cut-off function supported in the unit-ball of R d centered at 0 and such that, for all ξ ∈ R d ,…”